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A005846
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Primes of form n^2 + n + 41.
(Formerly M5273)
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40
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41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Note that 41 is the largest of Euler's Lucky numbers (A014556). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 22 2004
a(n)=A117530(13,n) for n<=13: a(1)=A117530(13,1)=A014556(6)=41, A117531(13)=13. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2006
The g.f. -(41-80*z+41*z**2)/(z-1)**3 conjectured by S. Plouffe in his 1992 dissertation is wrong.
The link to E. Wegrzynowski contents the following false statement: "It is possible to find a polynomial of the form n^2 + n + B that gives prime numbers for n = 0...A, A being any number." It is known that the maximum is A = 39 for B = 41. - Luis Rodriguez (luiroto(AT)yahoo.com), Jun 22 2008
Contrary to the last comment, Mollin's Theorem 2.1 shows that any A is possible if the Prime k-tuples Conjecture is assumed. [From T. D. Noe (noe(AT)sspectra.com), Aug 31 2009]
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REFERENCES
| R. A. Mollin, Prime producing quadratics, Amer. Math. Monthly 104 (1997), 529-544. [From T. D. Noe (noe(AT)sspectra.com), Aug 31 2009]
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Zak Seidov, Table of n, a(n) for n = 1..10000.
Phil Carmody, Drag Racing Prime Numbers! [from Vladimir Joseph Stephan Orlovsky, Jul 24 2011]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
E. Wegrzynowski, Les formules simples qui donnent des nombres premiers en grande quantite
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
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FORMULA
| a(n) =A056561(n)^2+A056561(n)+41
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EXAMPLE
| a(39)=1601=39^2+39+41 is in the sequence because it is prime. 1681=40^2+40+41 is not because 1681=41*41.
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MATHEMATICA
| Select[Table[n^2 + n + 41, {n, 0, 59}], PrimeQ] (* Alonso del Arte, Dec 08 2011 *)
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PROG
| (PARI) for(n=1, 1e3, if(isprime(k=n^2+n+41), print1(k", "))) \\ Charles R Greathouse IV, Jul 25 2011
(Haskell)
a005846 n = a005846_list !! (n-1)
a005846_list = filter ((== 1) . a010051) a202018_list
-- Reinhard Zumkeller, Dec 09 2011
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CROSSREFS
| Cf. A048988, A007634, A056561, A002378, A007635.
Intersection of A000040 and A202018; A010051.
Sequence in context: A054057 A155884 A202018 * A154498 A062669 A045710
Adjacent sequences: A005843 A005844 A005845 * A005847 A005848 A005849
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Henry Bottomley (se16(AT)btinternet.com), Jun 26 2000
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